Let's solve the problem step by step: --- ### **Step 1: Find a maximum (r, s)-flow** We need to find the maximum flow from **r** (source) to **s** (sink). #### **Capacities:** - r → a: 7 - r → b: 3 - a → d: 5 - b → a: 3 - b → c: 4 - d → b: 3 - d → s: 7 - c → d: 3 - c → s: 7 #### **Let's try sending as much flow as possible:** 1. **Path 1:** r → a → d → s - Minimum capacity along this path: min(7, 5, 7) = **5** - Send 5 units along this path. 2. **Path 2:** r → b → c → s - Capacity: min(3, 4, 7) = **3** - Send 3 units along this path. 3. **Path 3:** r → b → a → d → s - r → b: Already used full 3 units, so no capacity left. - This path is blocked. 4. **Path 4:** r → a → d → b → c → s - a → d: Already used 5/5 units, so no capacity left. - This path is blocked. 5. **Path 5:** r → a → d → b → c → d → s - Same as above, blocked at a → d. 6. **Path 6:** r → b → c → d → s - r → b: 3/3 used, so blocked. #### **Total Flow:** - **5 (via r-a-d-s) + 3 (via r-b-c-s) = 8** **At the end of Step 1:** **Maximum (r, s)-flow is 8.** --- ### **Step 2: Find a minimum (r, s)-cut** A cut separates the vertices into two sets, with **r** in one and **s** in the other, and the sum of the capacities of edges crossing from the r-side to the s-side is minimized. #### **Let's try different cuts:** 1. **Cut 1:** {r, a, b, c, d} | {s} - Edges crossing: d → s (7), c → s (7) - Capacity: 7 + 7 = 14 2. **Cut 2:** {r, a, b, c} | {d, s} - Edges crossing: a → d (5), c → d (3) - Capacity: 5 + 3 = 8 3. **Cut 3:** {r, a, b} | {c, d, s} - Edges crossing: b → c (4), a → d (5) - Capacity: 4 + 5 = 9 4. **Cut 4:** {r, a} | {b, c, d, s} - Edges crossing: r → b (3), a → d (5) - Capacity: 3 + 5 = 8 5. **Cut 5:** {r} | {a, b, c, d, s} - Edges crossing: r → a (7), r → b (3) - Capacity: 7 + 3 = 10 The **minimum cut** is **8**, which matches the maximum flow found above. - For example, **encircle {r, a, b, c}** as the subset for the minimum cut. **At the end of Step 2:** **The minimum (r, s)-cut has capacity 8, matching the max flow.** --- ## **Final Answer:** - **Maximum (r, s)-flow is 8.** - **Minimum (r, s)-cut has capacity 8** (e.g., by encircling {r, a, b, c}). - This proves the flow is maximum by the **Max-Flow Min-Cut Theorem**. --- **Let me know if you want the flow assignments for each edge or a marked-up image!**

Let's proceed step by step to solve parts (b) and (c): --- ### **(b) Find a maximum (r, s)-flow** **Step 1: Identify augmenting paths and send flow** Given the network capacities: - r → a: 7 - r → b: 3 - a → d: 5 - b → a: 3 - b → c: 4 - d → b: 3 - d → s: 7 - c → d: 3 - c → s: 7 **Initial flow assignment:** 1. **Path 1:** r → a → d → s - Capacity limited by min(7, 5, 7) = **5** - Send 5 units along this path. 2. **Path 2:** r → b → c → s - Capacity limited by min(3, 4, 7) = **3** - Send 3 units along this path. **Total flow:** 5 + 3 = **8** No more augmenting paths with available residual capacity exist, so the maximum flow from r to s is **8**. --- ### **(c) Prove the flow is maximum by identifying a minimum (r, s)-cut** **Step 2: Find the minimum cut** Possible cuts: - **Cut A:** {r, a, b, c} | {d, s} - Crossing edges: a → d (5), c → d (3) - Capacity = 5 + 3 = **8** - **Cut B:** {r, a, b, c, d} | {s} - Crossing edges: d → s (7), c → s (7) - Capacity = 7 + 7 = 14 - Other cuts have capacities ≥8. **The minimum cut capacity is 8**, matching our maximum flow. **Therefore,** by the Max-Flow Min-Cut Theorem, the maximum flow is **8**, and the minimum cut capacity is **8**. --- ### **Final answer:** - **Maximum (r, s)-flow:** **8** - **Minimum (r, s)-cut capacity:** **8** (e.g., the cut separating {r, a, b, c} from {d, s}) This confirms the flow is maximum, as the maximum flow equals the capacity of the minimum cut.